Abstract: This conservative averaging method was developed 30 years ago in my doctor dissertation. Now I and my collaborators had generalized this method for Descartes’, cylindrical and spherical system of coordinates.
By modeling practically interesting processes, very often we need to consider the situation, when the medium has a layered structure. Speaking mathematically, such situation can be described by PDE (or its system) with piecewise constant/continuous coefficients. In this lecture I’ll show how conjugations conditions, non-ideal contact conditions etc. and their generalizations can be obtained by our original method of conservative averaging (CAM). The usage of CAM for separate relatively thin sub-domain or/and for sub-domain with large medium characteristic, leads to reduction of domain, in which the solution must be found. To apply CAM procedure for several layers, it is necessary to construct special type of polynomial and rational spline, which interpolates the integral averaged values of the solution. New type of spline was developed for non-continuous solutions. In this case the original problem with discontinuous coefficients from transforms to problem with continuous coefficients in . The usage of CAM for ill-posed inverse problems in some cases transforms them to well-posed inverse problems.
We demonstrate this method for linear, non-linear heat transfer problems: convective conductive stream in layered system, intensive steel quenching, heat exchange in system with extended surface and electrical fuse.

Brief Biography of the Speaker:
Andris Buikis received the M.S. in numerical mathematics from University of Latvia (Faculty of Physics and Mathematics) in 1963 and Dr.math. (Candidate of Science in former USSR), University of Latvia, 1970. He was Junior Researcher, Senior Researcher, Computing Centre, University of Latvia, 1962 – 1972. Assistant Professor and Head of Chair of Applied Mathematics, 1972 – 1976 and Head of Chair of Differential Equations and Numerical Methods, Faculty of Physics and Mathematics, University of Latvia, 1976 – 1984. Dr.habil.math. (Doctor of Science in former USSR), University of Kasan, Russia, 1988. Professor, University of Latvia, 1991. Director, Institute of Mathematics, Latvian Academy of Sciences and Latvian University, 1991 - 1996; 2003 – 2006 and Director, Science and Dialogue Centre of Latvia, 1993 -2007. Head of Laboratory of Mathematical Technologies, Institute of Mathematics and Computer Science, University of Latvia 2006-2010.
In 2010 he was elected as Member of Saeima (Latvian Parliament). Full Member, Latvian Academy of Sciences, 1997. Member, Senate of the Latvian Academy of Sciences, 1994 –. The Latvian Academy of Sciences Piers Bohl Prize for a cycle of papers “Method of Conservative Averaging, Theory and Applications”, 2005. Member of Editorial Advisory Board, Journal Mathematical Modelling and Analysis (The Baltic Journal on Mathematical Applications, Numerical Analysis and Differential Equations), Lithuania 1999- and Associate Editor of WSEAS Transactions in Heat and Mass Transfer 2007-.